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**Conformal structures on compact complex manifolds**

### Nicolina Istrati

**To cite this version:**

Nicolina Istrati. Conformal structures on compact complex manifolds. General Mathematics [math.GM]. Université Sorbonne Paris Cité, 2018. English. �NNT : 2018USPCC054�. �tel-02156198�

### de l’Université Sorbonne Paris Cité

### Préparée à l’Université Paris Diderot

**École doctorale de sciences mathématiques de Paris centre**

### Discipline : Mathématiques

présentée par

**Nicolina Istrati**

**Conformal structures on compact complex**

**manifolds**

### dirigée par Andrei Moroianu

### Soutenue le 15 juin 2018 devant le jury composé de :

M. Paul Gauduchon Directeur de recherche École Polytechnique Président du jury
M. Paul Laurain Maitre de conférences Université Paris Diderot Examinateur
M. Andrei Moroianu Directeur de recherche Université Paris - Sud Directeur
Mme _{Mihaela Pilca} _{Professeur} _{Universität Regensburg} _{Examinatrice}
M. Uwe Semmelmann Professeur Universität Stuttgart Examinateur

### Au vu des rapports de :

M. Andrew Swann Professeur Aarhus University

Institut de mathématiques de Jussieu-Paris Rive gauche. UMR 7586.

Boîte courrier 247 4 place Jussieu 75 252 Paris Cedex 05

Université Sorbonne Paris Cité. Campus Paris Diderot.

École doctorale de sciences mathématiques de Paris centre. Case courrier 7012

8 Place Aurélie Nemours 75 205 Paris cedex 13

Je dois commencer par remercier mon directeur de thèse, Andrei Moroianu, qui a accepté dès mon M2 de surveiller et guider mes débuts dans la recherche. De m’avoir consacré énormément de temps et de patience en m’écoutant, en lisant mes nombreux essais de preuves et en m’indiquant des pistes de recherche. D’avoir corrigé beaucoup de mes erreurs et m’avoir constamment rappelé l’importance de l’esprit critique pour faire des mathématiques. Je remercie aussi Simone Diverio, qui nous a accompagnés pendant la première partie de cette aventure.

Ensuite je remercie Vestislav Apostolov et Andrew Swann d’avoir accepté d’être rapporteurs de ce manuscrit, ainsi que pour leurs remarques avisées. Merci à Paul Gauduchon, Paul Laurain, Mihaela Pilca et Uwe Semmelmann de faire partie du jury, leur présence me fait grand honneur.

Mes remerciements vont aussi à Liviu Ornea, qui a été mon mentor à Bucarest, et sur le soutien duquel j’ai toujours pu compter. Merci encore à Victor Vuletescu et à Sergiu Moroianu pour leurs conseils et pour avoir veillé à mon parcours. Finalement, merci à Aurel Bârsan, qui a été le premier professeur à me faire prendre les mathématiques au sérieux.

Merci à mon amie mathématique, Alexandra. C’est une grande chance qu’on puisse partager nos dilemmes pour en faire des articles. Merci aussi à Cristina, qu’on a accompagnée vers la topologie, mais à la ﬁn seule elle y a survécu.

Je suis reconnaissante envers mes camarades du couloir qui ont fait que l’expérience de thésard à Jussieu soit aussi sympathique. Premièrement au petit groupe Macarena, Louis, Léo et Adrien B, avec lesquels au début je me sentais bien sans trop comprendre ce qu’ils disaient, mais avec les bières, les belotes et les balades espagnoles, ça a ﬁni par arriver. À eux se rajoutent les autres collègues de génération Jesua, Johannes, Adrien S, Arthur, Amiel, Cyrus. Ensuite à ceux qui nous ont si bien accueillis et appris — Liana, Maylis, Juliette, François, Olivier, Thibaut, Amine, Anne, John, Malick, Miguel, Ramanujan, Lucas, Valentin, Arthur-César, Joaquin, Hsueh-Yung. Et ﬁnalement à ceux qui sont venus après et prennent ﬁèrement le relais — Hugo, Thomas, Justin, Xavier, Mathieu, Eckhard, Vincent, Michou, Malo, Peiyi, Linyuan, Benoit.

Aussi importants ont été mes colocataires le long de ces quatre années. Merci à Apple qui m’a accueillie avec autant de chaleur et de générosité. Ensuite merci à la troupe d’Ivry, où il faut encore nommer Hanane et Fede, pour leur bonne humeur et leurs repas délicieux.

Les savoir là, surtout quand je revenais pour une bouﬀée d’air roumain, a aussi contribué à mener à bonne ﬁn cette aventure. Merci à Tatiana et Buga Chan, à Irina et Marian, à Anca et

ii

Sergiu. Mais surtout merci à Ioana, sa compréhension et son encouragement ont été essentiels pour que je continue.

Je suis fortunée d’avoir rencontré Louis au milieu de tous ces mathématiques, partager ces dernières années ensemble a été une quête au moins aussi intéressante que le travail pour cette thèse. Je lui suis reconnaissante pour le savoir à mes côtés.

Finalement, je veux remercier ma famille. À mon frère, dont les propres recherches ont toujours été une inspiration pour moi. Et à mes parents, qui m’ont oﬀert leur soutien et la liberté de marcher dans mes propres pas. Merci.

Dans cette thèse on s’intéresse à deux types de structures conformes non-dégénérées sur une variété complexe compacte donnée. La première c’est une forme holomorphe symplectique twistée (THS), i.e. une deux-forme holomorphe non-dégénérée à valeurs dans un ﬁbré en droites. Dans le deuxième contexte, il s’agit des métriques localement conformément kähleriennes (LCK).

Dans la première partie, on se place sur un variété de type Kähler. Les formes THS généralisent les formes holomorphes symplectiques, dont l’existence équivaut à ce que la variété admet une structure hyperkählerienne, par un théorème de Beauville. On montre un résultat similaire dans le cas twisté, plus précisément: une variété compacte de type kählerien qui admet une structure THS est un quotient ﬁni cyclique d’une variété hyperkählerienne. De plus, on étudie sous quelles conditions une variété localement hyperkählerienne admet une structure THS. Dans la deuxième partie, les variétés sont supposées de type non-kählerien. Nous présentons quelques critères pour l’existence ou non-existence de métriques LCK spéciales, en terme du groupe de biholomorphismes de la variété. En outre, on étudie le problème d’irréductibilité analytique des variétés LCK, ainsi que l’irréductibilité de la connexion de Weyl associée. Dans un troisième temps, nous étudions les variétés LCK toriques, qui peuvent être déﬁnies en analogie avec les variétés de Kähler toriques. Nous montrons qu’une variété LCK torique compacte admet une métrique de Vaisman torique, ce qui mène à une classiﬁcation de ces variétés par le travail de Lerman.

Dans la dernière partie, on s’intéresse aux propriétés cohomologiques des variétés d’Oeljeklaus-Toma (OT). Plus précisément, nous calculons leur cohomologie de de Rham et celle twistée. De plus, on démontre qu’il existe au plus une classe de de Rham qui représente la forme de Lee d’une métrique LCK sur un variété OT. Finalement, on détermine toutes les classes de cohomologie twistée des métriques LCK sur ces variétés.

**Mots-clés**

Forme holomorphe symplectique, variété hyperkählerienne, métrique localement conformément kählerienne, métrique de Vaisman, géométrie torique, variété d’Oeljeklaus-Toma, cohomologie twistée.

**Abstract**

In this thesis, we are concerned with two types of non-degenerate conformal structures on a given compact complex manifold. The ﬁrst structure we are interested in is a twisted holomorphic symplectic (THS) form, i.e. a holomorphic non-degenerate two-form valued in a line bundle. In the second context, we study locally conformally Kähler (LCK) metrics. In the ﬁrst part, we deal with manifolds of Kähler type. THS forms generalise the well-known holomorphic symplectic forms, the existence of which is equivalent to the manifold admitting a hyperkähler structure, by a theorem of Beauville. We show a similar result in the twisted case, namely: a compact manifold of Kähler type admitting a THS structure is a ﬁnite cyclic quotient of a hyperkähler manifold. Moreover, we study under which conditions a locally hyperkähler manifold admits a THS structure.

In the second part, manifolds are supposed to be of non-Kähler type. We present a few criteria for the existence or non-existence for special LCK metrics, in terms of the group of biholomorphisms of the manifold. Moreover, we investigate the analytic irreducibility issue for LCK manifolds, as well as the irreducibility of the associated Weyl connection. Thirdly, we study toric LCK manifolds, which can be deﬁned in analogy with toric Kähler manifolds. We show that a compact toric LCK manifold always admits a toric Vaisman metric, which leads to a classiﬁcation of such manifolds by the work of Lerman.

In the last part, we study the cohomological properties of Oeljeklaus-Toma (OT) manifolds. Namely, we compute their de Rham and twisted cohomology. Moreover, we prove that there exists at most one de Rham class which represents the Lee form of an LCK metric on an OT manifold. Finally, we determine all the twisted cohomology classes of LCK metrics on these manifolds.

**Keywords**

Holomorphic symplectic form, hyperkähler manifold, locally conformally Kähler metric, Vais-man metric, toric geometry, Oeljeklaus-Toma Vais-manifold, twisted cohomology.

**Introduction** **vii**

Notation and conventions . . . xvi

**1 Twisted Holomorphic Symplectic Forms** **1**
1.1 Introduction . . . 1

1.2 Holomorphic symplectic manifolds . . . 2

1.3 Twisted holomorphic symplectic manifolds . . . 3

1.4 A characterization . . . 7

1.5 Examples . . . 12

**2** **Locally Conformally Kähler Geometry** **15**
2.1 Introduction . . . 15

2.2 Basic deﬁnitions and properties . . . 16

2.3 Connections . . . 19

2.4 Special LCK metrics . . . 22

2.5 Inﬁnitesimal automorphisms of LCK manifolds . . . 25

2.6 Examples . . . 28

2.6.1 Diagonal Hopf manifolds . . . 28

2.6.2 Non-diagonal Hopf surfaces . . . 31

2.6.3 LCK manifolds obtained from ample vector bundles . . . 32

2.6.4 LCK metrics on blow-ups . . . 34

2.6.5 Complex surfaces . . . 35

**3 Existence Criteria for LCK Metrics** **37**
3.1 Introduction . . . 37

3.2 The Lee vector ﬁeld . . . 38

3.3 Existence of LCK metrics with potential . . . 42

3.4 Existence of Vaisman metrics . . . 44

3.5 Torus principal bundles . . . 47

3.6 Analytic irreducibility of complex manifolds of LCK type . . . 48

3.7 Weyl reducible manifolds . . . 51 v

**4 Toric LCK Manifolds** **55**

4.1 Introduction . . . 55

4.2 Twisted Hamiltonian Vector Fields . . . 56

4.3 Torus actions on LCS manifolds . . . 58

4.4 Proof of the Main Theorem . . . 59

4.5 Examples . . . 63

4.6 Final remarks and questions . . . 65

**5 Cohomological properties of OT manifolds** **67**
5.1 Introduction . . . 67

5.2 Oeljeklaus-Toma manifolds . . . 68

5.2.1 The construction . . . 68

5.2.2 Metric properties . . . 69

5.3 Technical Preliminaries . . . 71

5.3.1 Leray-Serre spectral sequence of a locally trivial ﬁbration . . . 71

5.3.2 Twisted cohomology . . . 73

5.4 The de Rham cohomology . . . 76

5.5 The Leray-Serre spectral sequence of OT manifolds . . . 79

5.6 Twisted cohomology of OT manifolds . . . 81

5.7 Applications and Examples . . . 83

In the present dissertation, we are interested in certain non-degenerate conformal structures
on a given compact complex manifold. The conformal nature can be encoded in a line bundle
*over the manifold. As such, given a complex manifold (M, J) and a line bundle L over M, we*
want to study non-degenerate two-forms:

*ω*∈ Γ(V2*T*∗*M⊗ L).*

*There are two diﬀerent settings which we investigate. In the ﬁrst one, we suppose that L is a*
*holomorphic line bundle and ω is an L-valued holomorphic two-form. This kind of structure*
*will be called twisted holomorphic symplectic (THS). In the second context, we suppose that L*
*is an oriented real line bundle endowed with a ﬂat connection ∇, and ω is a positive (1, 1)-form*
*with values in L which is d*∇*-closed. Such a structure will be called a locally conformally*

*Kähler form (LCK).*

Both these structures are natural generalisations of the well-known non-twisted ones: the ﬁrst
*one coincides with a holomorphic symplectic form when L is holomorphically trivial, while the*
*second one is simply a Kähler metric when (L, ∇) = (M × R, d). We wish to understand what*
kind of restrictions the existence of such structures imposes on the manifold, and to what
extent the properties of the corresponding non-twisted structures generalise to our setting.
*As it turns out, if one assumes that (M, J) is of Kähler type, then one reduces quite easily to*
the non-twisted situation. The ﬁrst chapter proves and explains this reduction in the context
of THS structures. On the other hand, we make no assumption of Kählerness in the second
situation, and although LCK structures are just conformal generalisations of Kähler structures,
they behave quite diﬀerently from the latter. LCK structures are studied throughout chapters
2 to 4. The last chapter presents a certain family of non-Kähler complex manifolds, called
Oeljeklaus-Toma manifolds, focusing on their topological properties. This part is related to
the rest of the discussion by the fact that some of these manifolds admit LCK forms.

**Twisted holomorphic symplectic forms**

*Let (M, J) be a compact complex manifold of Kähler type, of complex dimension 2m. It is*
well known, by a theorem of Beauville [Bea83b] based on Yau’s proof of the Calabi conjecture,
*that the existence of a holomorphic symplectic form ω on M is equivalent to the manifold*
admitting a hyperkähler structure, i.e. a metric compatible with the complex structure, whose
*holonomy group of the Levi-Civita connection sits in Sp(m). One might hope to obtain less*
rigid structures if one assumes that the symplectic form takes values in a holomorphic line
bundle instead.

viii

This expectation is quite natural if we take a look at the analogous symmetric situation:
*suppose that L is a holomorphic line bundle over (M, J) and that there exists a non-degenerate*
*holomorphic section g ∈ H*0* _{(M, S}*2

*∗*

_{T}

_{M}_{⊗ L), called a holomorphic conformal structure. Still}*under the Kählerness assumption, Inoue, Kobayashi and Ochiai [IKO80] proved that if L is*

*holomorphically trivial then (M, J) is a ﬁnite quotient of a complex torus. On the other hand,*

*if L is not trivial, then new examples appear, and in fact classiﬁcations are known only for*the compact surfaces ([KO82]), and for projective threefolds ([JR05]). Let us note that the standard example of such a manifold is given by the hyperquadric:

Q*n= {[z*0 *: · · · : zn+1]| − 2z*0*zn+1*+
*n*

X

*k=1*

*z*2* _{k}*= 0} ⊂ P

*n*

*with the structure g = −2dz*0*dzn+1+ dz*1*dz*1*+ . . . dzndzn∈ H*0(Q*n, S*2*T*∗Q*n⊗ O(2)).*

It turns out that things are diﬀerent in the symplectic setting, as we show that the conformal case is quite similar to the standard one:

**Theorem A** **(Theorem 1.3.5, Theorem 1.4.1). Let (M**2m_{, J, L, ω), m > 1, be a compact THS}

*manifold of Kähler type, and let α∈ H*2_{(M, R) be a Kähler class. Then L is unitarily flat, and}

*there exists a unique Kähler metric g with respect to J representing α so that a finite cyclic*
*cover of (M, g, J) has holonomy in Sp(m). Moreover, the form ω is parallel with respect to*
*the natural connection induced by g on* V2*T*∗*M* *⊗ L.*

*The main point of the proof is to show that the line bundle L has torsion ﬁrst Chern class, as*
everything else follows similarly to the non-twisted case, via Yau’s theorem and the Weitzenböck
*formula. This is true, because we manage to construct a holomorphic connection in L, naturally*
*induced by ω via the Lefschetz operator*

Lef*ω* : Ω• → Ω•+2*⊗ L, η 7→ η ∧ ω.*

*By the Kähler assumption, this will imply then that c*1*(L) = 0 ∈ H*2*(M, R).*

*Let us note that the hypothesis m > 1 is natural, as any complex surface admits a THS form:*
*simply take L =*V2* _{T M , and note that Ω}*2

*M⊗ L is holomorphically trivial.*

*An equivalent deﬁnition for a hyperkähler structure on M is a Riemannian metric g together*
*with three integrable complex structures I, J, K compatible with g, parallel for the Levi-Civita*
*connection of g and verifying the quaternionic relations IJ = K = −JI. Moreover, we say*
*that (M, g) is locally hyperkähler if the universal Riemann cover ( ˜M , ˜g) is hyperkähler, so*

*that the structures I, J and K are deﬁned only locally on M. Note that the above theorem*
*gives the existence of a local hyperkähler structure on M which is particular: it is formed*
*of a global complex structure J which we have ﬁxed, together with two more local complex*
*structures. We call such a structure Kähler locally hyperkähler (KLH).*

In the last part of Chapter 1, we investigate under what conditions a KLH manifold admits a twisted holomorphic symplectic form. The presence of a THS structure forces the fundamental group of the manifold to have a certain structure, which we describe. This depends mainly on the (local) de Rham reducibility of the manifold. We ﬁrst note that, although a product of two hyperkähler manifolds is again hyperkähler, THS manifolds are de Rham irreducible (Corollary 1.4.2). At the same time, we show that for locally irreducible manifolds, the existence of a THS structure is equivalent to the manifold being KLH (Corollary 1.4.3). For the intermediate

case of irreducible, locally reducible manifolds, we need to do a discussion depending on the ﬁniteness of the fundamental group. The results of this part are obtained by an analysis of the structure of an isometry of certain Riemannian products, and the main tool we use is the holomorphic Lefschetz ﬁxed-point formula.

**Locally conformally Kähler metrics**

*A Kähler metric g on a complex manifold (M, J) is a Hermitian metric whose fundamental*
*form Ω := g(J·, ·) is closed. There are many well-known obstructions to the existence of such*
*metrics on a compact manifold, the basic one being that b*1 needs to be even. One way to
generalize such metrics is to consider metrics that are conformal to them. It can be easily
*seen that if Ω is Kähler and f ∈ C*∞_{(M, R), then e}f_{Ω will not be Kähler unless f is constant.}

More generally, in order to get rid of some of the topological obstructions, one can consider
metrics that are only locally conformal to Kähler metrics: these are the LCK metrics. More
*precisely, Ω is LCK if every point of M has a neighbourhood U on which there exists a Kähler*
metric Ω*U* which is conformal to Ω, i.e.

Ω|*U* = e*fU*Ω*U,* *fU* ∈ C∞*(U, R).* (0.0.1)

*In the above equation, although the function fU* *is local, θ = dfU* is a global real closed

*one-form on M, called the Lee form of Ω, provided that dim*C*M > 1. This allows us to give*
*an equivalent deﬁnition: g is LCK if Ω veriﬁes dΩ = θ ∧ Ω, with θ a closed one-form on M.*
On the minimal cover ˆ*M of M on which θ becomes exact, the local Kähler metric in (0.0.1)*

pulls back to a global Kähler metric Ω*K*, and ( ˆ*M , J, ΩK) is called the minimal Kähler cover.*

The most basic manifold of non-Kähler type admits an LCK metric: this is the (standard) Hopf surface

*H = C*2_{− {0}/}_{(z}_{1}_{,z}_{2}_{)∼(αz}_{1}_{,αz}_{2}_{)}*,* *0 < α < 1,*

*with the Boothby metric Ω = |z|*−2* _{dd}c_{|z|}*2

_{. This is also the ﬁrst LCK example appearing in}the literature, but although the metric was constructed by Boothby in [Bo54], it was noticed only after twenty years that it is LCK by Vaisman, who was the one to start a systematic study of these structures.

Any LCK metric on a manifold of Kähler type is globally conformal to a Kähler metric
([Va80]). For this reason, we will always assume tacitly that our manifolds are not of Kähler
type, in order to study only strict LCK metrics. In this setting, a ﬁrst obstruction appears
*for manifolds of LCK type, namely: 0 < b*1 *< 2h0,1, where h0,1* = dimC*H*1*(M, OM*). As a

matter of fact, this is the only cohomological obstruction known for a general LCK manifold.
*Vaisman had conjectured that such a manifold should always have b2k+1odd for some k ∈ N,*
however this was disproved by the OT manifolds [OT05].

There are a few special LCK metrics which are better understood. The most important one is a

*Vaisman metric (which used to be called generalized Hopf), deﬁned by the condition* _{∇}*g _{θ = 0,}*

where ∇*g* _{is the Levi-Civita connection determined by g. Their study started with Vaisman}

([Va82]) and is still ongoing. Vaisman manifolds admit a transversal Kähler foliation, which allows one to study some of their properties by means of the better known ones from Kähler geometry. For instance, this was done in order to determine their cohomological properties by Vaisman [Va82] and Tsukada [Ts94], and it turns out that the Frölicher spectral sequence degenerates at the ﬁrst page for manifolds of Vaisman type (although they don’t admit a Hodge decomposition, as already noticed).

x

*Moreover, a normalised Vaisman metric (Ω, θ) on (Mn _{, J) has the form}*

*Ω = −dJθ + θ ∧ Jθ,* (0.0.2)

(see Corollary 2.4.8 in Chapter 2), and the corresponding Kähler metric on ˆ*M can be written*

as Ω*K* *= ddc*e*−ϕ* *where ϕ ∈ C*∞( ˆ*M , R) is a function satisfying θ = dϕ on ˆM . Thus ΩK*

has a positive potential. This was ﬁrst noted by Verbitsky [Ve04], and as a consequence Ornea-Verbitsky [OV10] introduced and started the study of the more general notion of a

*LCK metric with (positive) potential. These are LCK metrics whose Kähler metric writes*

Ω*K* *= ddc(fe−ϕ),* *f* ∈ C∞*(M, R).* (0.0.3)

This class of metrics has the advantage of being closed under small deformations ([OV10],
[Go14]), while the Vaisman manifolds are not (see [Bel00]). Even more general than this is the
*notion of an exact LCK metric, which is an LCK metric whose Kähler metric has the form:*

Ω*K* *= d(e−ϕη),* *η*∈ E1*(M, R).* (0.0.4)

*A closed one-form θ on a manifold M induces a twisted diﬀerential operator*

*dθ*: E*k*→ E*k+1,* *α7→ dα − θ ∧ α*

*which veriﬁes d*2

*θ* *= 0. This deﬁnes the twisted cohomology Hθ(M) = Ker dθ/ Im dθ*, which

*plays an important role in LCK geometry. Note that an LCK structure (Ω, θ) induces a*
*cohomology a class [Ω] ∈ H*2

*θ(M), which is zero precisely for exact LCK metrics. Moreover, by*

*[LLMP03], one has H*•

*θ(M) = 0 if θ is the Lee form of a Vaisman metric.*

Finally, let us say a few words on an interesting problem in LCK geometry, which we also tackle in the special case of OT manifolds. It concerns determining the set, or at least the geometry of the set:

*L(M, J) = {a ∈ H*1*(M, R)| there exists an LCK structure (Ω, θ) with θ ∈ a}*

*where (M, J) is a compact complex manifold. It was ﬁrst studied by Tsukada in [Ts94]*
*in the case of a Vaisman manifold (M, J). He showed that any element of L(M, J) is*
*the Lee class of a Vaisman metric, and that for a given element a*0 *∈ L(M, J), one has*
*L(M, J) = {ta*0*+ b|t > 0, b ∈ H ⊂ H*1*(M, R)}. Here, H ⊂ H*1*(M, R) is formed by all the de*
*Rham classes of forms whose (1, 0)-part are holomorphic d-closed one forms. Tsukada showed*
*that on a Vaisman manifold, H is a hyperplane in H*1* _{(M, R). More recently, the set L(M, J)}*
was studied by Apostolov and Dloussky [AD16a], [AD16b] and by Otiman [O16] in the case of
compact complex surfaces, and it has been completely determined for a number of cases. By
the above cited articles, together with our result for OT manifolds, it turns out that for all the

*known examples of LCK manifolds, L(M, J) is either a point or an open subset of H*1

_{(M, R).}**Existence of LCK metrics**

The ﬁrst question one probably asks is how often do LCK metrics arise? As noted, simply connected compact complex manifolds of non-Kähler type, such as Calabi-Eckmann manifolds for instance, cannot admit such metrics, so the class is strict. The existence of LCK metrics on compact complex surfaces is fairly well undestood ([Tr82], [LeB91], [GO98], [Bel00], [FP10],

[Bru11]), and the only ones known to not admit LCK metrics are a certain class of Inoue-Bombieri surfaces. The surfaces of this class are in fact small deformations of some other Inoue-Bombieri surfaces which admit LCK metrics, so in particular the category of LCK manifolds is not closed under small deformations. Moreover, one encounters all the special LCK metrics deﬁned above, or the lack of such metrics, already in the surface case.

Moving to higher dimension, there are natural ways of constructing Vaisman manifolds: starting from any ample holomorphic vector bundle over a Kähler manifold, one has a naturally associated Vaisman manifold ([Va76], [Va80], [Ts97], [Ts99]). We extend this construction in Section 2.6.3 to obtain manifolds with LCK metrics with positive potential. Moreover, any complex submanifold of dimension bigger than 1 of a Vaisman manifold is again Vaisman ([Va82], [Ts97]). However, it is more diﬃcult to construct examples of manifolds of LCK type, not admitting LCK metrics with potential (or exact, for that matter). In fact, all known manifolds of higher dimension of this kind are either Oeljeklaus-Toma manifolds, or blow-ups of LCK manifolds. We should note at this point that indeed, the blow-up of an LCK manifold along a submanifold of Kähler type admits an LCK metric ([Tr82], [Vu09], [OVV13]), but never an exact one.

It should also be noted that the product metric of two LCK manifolds cannot be LCK [Va80],
however it is still unknown whether such a product manifold can admit some other LCK metric.
*Some particular cases are known, such as: if M*1 *and M*2 are of Vaisman type, then their
*product admits no LCK metric [Ts99], and if M*1 *is not a curve and veriﬁes the ∂ ¯∂-lemma,*
*then again M*1*× M*2 admits no LCK metric [OPV14]. We moreover prove:

**Theorem B** * (Theorem 3.6.3, Proposition 3.6.4). Suppose that M*1

*and M*2

*are two compact*

*complex manifolds. Then M*1*× M*2 *cannot admit a Vaisman metric. Moreover, if M*1 *is of*

*Vaisman type, then M*1*× M*2 *admits no LCK metric at all.*

**Theorem C** * (Proposition 3.6.8). Let M*1

*be a compact complex curve, let M*2

*be a complex*

*manifold and suppose that M := M*1*× M*2 *admits an LCK metric. Then M*2 *admits an LCK*

*metric with positive potential.*

A related problem concerns the reducibility of the natural connections associated to an LCK
metric. Madani, Moroianu, and Pilca showed in [MMP16] that the holonomy group of the
Levi-Civita connection of an LCK metric is irreducible and generic, unless the metric is
*Vaisman, in which case it equals SO(2n − 1), n being the complex dimension of the manifold.*
*Another natural connection associated to an LCK metric (Ω, θ) on (M, J) is the standard*
*Weyl connection D, deﬁned as being the unique torsion free connection on M which satisﬁes:*

*DJ = 0, DΩ = θ ⊗ Ω.*

As this connection coincides with the Levi-Civita (or also the Chern) connection of the local Kähler metrics, it encodes in some sense more interesting properties. For instance, the Weyl connection of an LCK metric can have reducible holonomy in a non-trivial way: this is the case for the Oeljeklaus-Toma manifolds. Kourganoﬀ [Kou15] gave a structure theorem for a more general class of manifolds with Weyl-reducible connection, which we adapt to the LCK context, in order to show:

**Theorem D** **(Theorem 3.7.7). Any exact LCK metric on a compact complex manifold is**

xii

The fact that we know very few examples of non-exact LCK metrics parallels with the lack of criteria for the existence or non-existence of a general LCK metric. Again, things look better if we turn to the case of metrics with positive potential or of Vaisman metrics. Ornea-Verbitsky ([OV12], [OV17]) gave an existence criterion for LCK metrics with positive potential, whose

proof we revise:

**Theorem E ([OV12], [OV17], Theorem 3.3.1). Let (M, J, Ω, θ) be a compact LCK manifold**

*admitting a holomorphic action of S*1 _{which, on the minimal cover ˆ}_{M , lifts to an effective }

*R-action. Then there exists an LCK metric with positive potential whose Lee form is cohomologous*
*to θ.*

*Kamishima-Ornea [KO05] gave a criterion for a given LCK conformal class [g] on a compact*
*complex manifold (M, J) to admit a Vaisman metric, namely they show that this is equivalent*
*to the automorphism group Aut(M, J, [g]) containing a one-dimensional complex Lie group*
which does not act isometrically on the corresponding Kähler metric. We generalise their
criterion in a way that does not involve a given ﬁxed conformal class:

**Theorem F** **(Theorem 3.4.3). A connected compact complex manifold (M, J) of LCK type**

*admits a Vaisman metric if and only if Aut(M, J) contains a torus T whose Lie algebra t*
*verifies dim*C(t*∩ it) > 0.*

As a consequence, we obtain a criterion of non-existence of LCK metrics:

**Corollary G (Corollary 3.4.5). Let (M, J) be a compact complex manifold, and suppose that**

*the group of biholomorphisms Aut(M, J) contains a compact torus whose Lie algebra t verifies*

dimC(t*∩ it) > 1. Then (M, J) admits no LCK metric.*

An immediate application of this criterion is the classiﬁcation of manifolds of LCK type among all the torus principal bundles (Proposition 3.5.1), in analogy with a theorem of Blanchard [Bl54].

*Let us recall at this point that an LCK metric (Ω, θ) induces naturally two vector ﬁelds B*
*and A = JB, called the Lee and Reeb vector ﬁelds, via:*

*ιA*Ω =*−θ, ιBΩ = Jθ.*

*In the case of a Vaisman metric, these vector ﬁelds are part of the Lie algebra aut(M, J, Ω),*
and are the ones to generate a non-real torus as in the above criterion. Moreover, the condition
*of B being Killing easily implies that the metric is Vaisman. One could then ask what happens*
*if we impose B to be holomorphic. This problem is studied in the recent paper [MMO17],*
where A. Moroianu, S. Moroianu and L. Ornea show that if, moreover, the LCK metric is
*Gauduchon, or if B has constant norm, then the metric is Vaisman. At the same time, they*
construct an example of a non-Vaisman LCK metric with holomorphic Lee ﬁeld, showing
*that one needs more hypothesis then just the holomorphicity of B. In Proposition 3.2.2 we*
*show that if Ω is an LCK metric with constant potential, i.e. of the form (0.0.2), and B is*
holomorphic, then again Ω is Vaisman. Moreover, we make the remark that the example of
[MMO17] can be chosen with positive potential (Lemma 3.2.7), so our hypothesis cannot be
relaxed.

**Toric LCK manifolds**

*Recall that a symplectic manifold (M2n _{, ω) of real dimension 2n is called a toric manifold if}*
the compact torus T

*n*

*acts eﬀectively in a Hamiltonian way on (M, ω). If t denotes the Lie*algebra of T

*n*, this means that there exists a T

*n-invariant map µ : M*

_{→ t}∗, called the moment

*map, verifying that for each vector ﬁeld X*

_{∈ t ⊂ aut(M), letting µ}X*=< µ, X >*∈ C∞

*(M ),*

*one has dµX* *= ιXω. As is well-known, by the Delzant construction, toric manifolds (M, ω, µ)*

are completely determined by the image of their moment map, which is a Delzant polytope.
Moreover, all their analytic and geometric properties can be read oﬀ their moment polytope,
*and in particular they admit a compatible complex structure with respect to which ω is Kähler.*
*If one forgets the complex structure in the LCK context, then one deals with locally conformally*

*symplectic (LCS) forms, namely non-degenerate forms Ω verifying dΩ = θ*_{∧ Ω for some closed}

*real one-form θ. For these structures, there exist analogous notions of Hamiltonians and*
*moment maps, introduced by Vaisman in [Va85]. A vector ﬁeld X on M is twisted Hamiltonian*
*with respect to (Ω, θ) if there exists a function fX* ∈ C∞*(M ) so that*

*ιXΩ = dθfX.* (0.0.5)

If we consider the associated minimal symplectic cover ( ˆ*M , ΩK*) as in the LCK context, then

*this deﬁnition is equivalent to asking for the lift of X to ˆM to be Hamiltonian for ΩK. If θ is*

*not exact and M is compact, then dθ* is injective on C∞*(M ), which implies that if a function*

as in (0.0.5) exists, then it is unique. Finally, let us note that this deﬁnition only depends on
*the conformal class [Ω], and not on Ω itself: X is twisted Hamiltonian for Ω if and only if it is*
so for e*fΩ, f* ∈ C∞_{(M, R).}

*A toric LCS manifold is an LCS manifold (M2n, [Ω]) together with the eﬀective action of*

a compact torus T*n* _{so that every induced vector ﬁeld X}_{∈ Lie(T}*n*_{)} _{⊂ aut(M) is twisted}

*Hamiltonian. If, moreover, there exists a complex structure on M so that Ω is LCK, and the*
*torus acts by biholomorphisms, then we have a toric LCK manifold. If µ is the moment map*
*of a toric LCS form (Ω, θ), then the minimal symplectic cover is a toric symplectic manifold*
with corresponding moment map ˆ*µ = e−ϕµ, where θ = dϕ.*

Although introduced early in the history of LCS/LCK geometry, general twisted Hamiltonian
group actions have not been extensively studied. The reduction procedure from symplectic
geometry has been adapted to this context by Haller and Rybicki in [HR01] for LCS manifolds,
and by Gini, Ornea and Parton in [GOP05] for LCK manifolds. Moreover, twisted Hamiltonian
actions were studied by Otiman in [O15] for the purpose of constructing LCS bundles.
On the other hand, recently there have been some new advances concerning toric LCK
manifolds. In order to explain them, let us ﬁrst recall that Vaisman metrics are closely related
to Sasaki structures: their minimal Kähler covers are Kähler cones over Sasaki manifolds, cf.
[GOP06]. Pilca showed in [Pi16] that a compact Vaisman manifold is toric if and only if the
associated Sasaki manifold is toric, and one action naturally induces the other. Moreover,
Madani, Moroianu and Pilca showed in [MMP17] that the ﬁrst Betti number of a toric Vaisman
*manifold is b*1= 1, implying that the associated toric Sasaki manifold is compact.

In the same paper, the authors gave a classiﬁcation of compact toric LCK surfaces, and it turns out that they all admit toric Vaisman metrics. Hence the question was raised of whether this is always the case, regardless of dimension. The main result of Chapter 4 is an aﬃrmative answer to it:

xiv

*exists a Vaisman metric Ω*′*, possibly nonconformal to Ω, with respect to which the same action*
*is still twisted Hamiltonian.*

Let us note that compact toric Sasaki manifolds can also be understood via the image of
the moment map, as a result of the paper [Ler03] of Lerman, in which he completes the
classiﬁcation of toric compact contact manifolds. Indeed, the image of their moment map to
which one adds_{{0} is a cone over a convex polytope with certain combinatorial properties}
*which makes it a good cone. Moreover, to each good cone one can associate in a unique way a*
*compact contact toric manifold (N, α). On the symplectic manifold naturally associated to*
this contact manifold, there always exists a compatible complex structure, inducing a toric
*Sasaki structure on N . Moreover, just like in the compact symplectic case, all such complex*
structures can be described only in terms of certain functions deﬁned on the moment cone, as
shown by Martelli, Sparks and Yau [MSY06], see also Abreu [Ab10].

With this in mind, and as a corollary of our result, we can thus describe also toric LCK manifolds in terms of combinatorial data coming from certain moment cones. However, all information is not preserved: from the good cones we can recover only some of the toric LCS structures, namely the ones giving Vaisman metrics.

We end this part by a remark on the diﬀerences between the symplectic case and the LCS case. Recall that a compact toric symplectic manifold admits a compatible integrable complex structure with respect to which the manifold is toric Kähler. On the contrary, Example 4.5.5 shows that on a general toric LCS manifold, there does not always exist a compatible complex structure, making it into a toric LCK manifold.

**Oeljeklaus-Toma manifolds**

OT manifolds were introduced by Oeljeklaus and Toma in [OT05] as higher dimensional
analogues of a class of Inoue-Bombieri surfaces. They are compact complex manifolds of
non-Kähler type, obtained as quotients of H*s*_{× C}*t*_{by discrete groups of aﬃne transformations}

*arising from a number ﬁeld K and a particular choice of a subgroup of units U of K. Usually,*
*such a manifold is said to be of type (s, t), and is denoted by X(K, U ).*

*More speciﬁcally, start with a number ﬁeld K which admits exactly s real embeddings in C,*

*σ*1*, . . . , σs, and 2t complex conjugate ones σs+1* *= σs+t+1, . . . , σs+t* *= σs+2t*. Then there exists

*a choice of a subgroup of units U of the ring of integers of K, OK*, so that the semi-direct

*product Γ := U ⋊ OK* acts freely and properly discontinuously on H*s*× C*t*, and the quotient

*X = X(K, U ) := Hs*× C*t _{/Γ is compact. Both groups O}*

*K* *and U act diagonally on Hs*× C*t*

*via the ﬁrst s + t embeddings: OK* *acts by translations, while U acts by dilatations.*

*OT manifolds of type (s, 1) are known to admit LCK metrics. But as they carry no holomorphic*
vector ﬁelds, they admit no Vaisman metrics. In fact, along with the blown-ups of LCK
manifolds, these are the only known examples of LCK manifolds in higher dimension which
admit no exact LCK metric, by a result of Otiman [O16]. Thus they are a good testing
ground for conjectures concerning cohomological properties of LCK manifolds. Indeed, when
introduced, they disproved a long standing conjecture of Vaisman, according to which the odd
index Betti numbers of an LCK manifold should be odd.

So far, signiﬁcant advances have been made in the study of OT manifolds. Many of their
*properties are closely related to the arithmetical properties of (K, U ), as can be seen particularly*

in the papers of M. Parton and V. Vuletescu [PV12] and of O. Braunling [Bra17]. OT manifolds
were shown to carry the structure of a solvmanifold by H. Kasuya [Kas13a], and those of type
*(s, 1) to contain no non-trivial complex submanifolds by L. Ornea and M. Verbitsky [OV11].*
A delicate issue seems to be the existence of LCK metrics on OT manifolds which are not
*of type (s, 1). Some progress in this direction has been made by V. Vuletescu [Vu14] and A.*
Dubickas [Du14], but the question remains open in general.

In the present text, we are interested in the cohomological properties of OT manifolds. Their
*ﬁrst Betti number and the second one for a certain subclass of manifolds, called of simple*

*type, were computed in [OT05]. More recently, H. Kasuya computed in [Kas13b] the de*

*Rham cohomology of OT manifolds of type (s, 1), using their solvmanifold structure. We*
will compute the de Rham cohomology algebra (Theorem 5.4.1) and the twisted cohomology
(Theorem 5.6.1) of any OT manifold. This is done in terms of numerical invariants coming

*from U* _{⊂ K.}

We do this by two diﬀerent approaches. In order to explain them, let us ﬁrst note the
diﬀerentiable ﬁber bundle structures appearing in the construction of an OT manifold. ˆ*X :=*

H*s*_{× C}*t/OK* has the structure of a trivial principal T*n*-bundle over R*s, where n = s + 2t. This*

*structure descends to X to a ﬂat Tn*_{-ﬁber bundle structure over T}*s*_{. Note that T}*n* _{acts on}

ˆ

*X, but not on X. Our ﬁrst approach consists in reducing to the study of the cohomology of*

T*n*-invariant diﬀerential forms. In the second one, we study the Leray-Serre spectral sequence
*associated to the ﬁber bundle structure of X, which turns out to degenerate at the second*
page.

In the rest of the chapter, we present a few applications, focusing on the OT manifolds of LCK type. First of all, we show:

**Theorem I (Proposition 5.2.2). Let X be an OT manifold of LCK type. Then X admits only**

*one Lee class.*

Next, we identify all the possible classes of LCK forms in the twisted cohomology group

*H*2

*θ(X, R) on an OT manifold of LCK type (Corollary 5.7.8). As a consequence of this, we*

*obtain that an LCK form (Ω, θ) on an OT manifold induces a non-degenerate Lefschetz map*
in cohomology, in the sense that LefΩ *: Hk(X, C)→ Hk+2(X, C) is injective for k*≤ dimC*X*
*and surjective for k*_{≥ dim}C*X.*

xvi

**Notation and conventions**

*• M, N will generally denote smooth manifolds.*

• ˜*M will always denote the universal cover of a manifold M .*

*• π*1*(M ) will be the fundamental group of M , and Γ will usually denote some normal*
*subgroup of π*1*(M ) (or even π*1*(M ) itself). These groups will be automatically identiﬁed*
*with the deck groups of the associated coverings of M .*

*• We will denote by capital letters the compact Lie groups, such as G, H etc, and by*
lowercase Gothic letters their corresponding Lie algebras, i.e. g, h etc.

*• g will denote a Riemannian metric on a given manifold.*

*• Connections will be denoted by D, ∇, D, and the curvature corresponding to a Chern*
connection, by Θ.

*• I, J, K will denote complex structures on a given manifold. If we ﬁx a complex structure*

*J on a smooth manifold M , then we will sometimes use the notation M also for the*

*complex manifold (M, J), when there is no ambiguity.*

*• KM* *will denote the canonical bundle of a given complex manifold (M, J).* O*M* will

*denote the sheaf of holomorphic functions of M .*
• Ω*k*

*M* *will denote the sheaf of holomorphic k-forms on a complex manifold (M, J), and*

E*Mp,q* *the sheaf of smooth (p, q)-forms.*

• E*k*

*M* *will denote the sheaf of real-valued smooth k-forms on M , and*E*Mk* ⊗ C the sheaf of

*smooth C-valued k-forms.*

*• Given a holomorphic vector bundle E over a complex manifold (M, J), H*0_{(M, E) will}*denote the holomorphic sections of E. Its corresponding smooth sections will be denoted*
by_{C}∞*(M, E) or by Γ(M, E). Also, by some abuse of notation, we will denote by*_{E}* _{M}p,q_{⊗E}*
or by

_{E}

_{M}p,q(E) the sheaf of (p, q)-forms on M valued in E.*• Let (L, h) be a Hermitian line bundle over (M, J), and let Θh* denote the curvature of

*the induced Chern connection of L. We use the convention that c*1*(L), the ﬁrst Chern*
*class of L, is the de Rham cohomology class of* * _{2π}i* Θ

*h. We will either view it in H*2

*(M, R)*

*or in H*2_{(M, Z).}

• T*n* _{denotes the n-dimensional compact torus, seen as a real Lie group. We will denote}

**by T a complex compact torus.**

*• For X ∈ Γ(T M) a smooth vector ﬁeld on a manifold M, ιX* denotes the contraction

*with X, while* _{L}*X* *denotes the Lie derivative with respect to X.*

In the context of Locally Conformally Kähler geometry:

*• Ω will denote the LCK form. θ ∈ E*1_{(M ) will denote the Lee form corresponding to Ω,}*verifying dΩ = θ*_{∧ Ω.}

• [Ω] will denote the conformal class of Ω, that is the set {e*f*_{Ω}* _{|f ∈ C}*∞

_{(M )}_{}, where M is}*the ambient manifold. Similarly, [g] will denote the conformal class of a Riemannian*

*metric g.*

*• B will denote the Lee vector ﬁeld corresponding to Ω, deﬁned by ιBΩ = Jθ, and*

*A the Reeb ﬁeld A = JB, also deﬁned by ιA*Ω = *−θ. Equivalently, if there exists a*

*compatible complex structure on the given manifold and g = Ω(·, J·) is the corresponding*
*Riemannian metric, then B and A are the duals of θ and Jθ with respect to g.*

• ˆ*M will denote the minimal cover of (M, θ) on which θ becomes exact. ϕ will be a*

function on ˆ*M or on ˜M satisfying dϕ = θ, and ΩK* will denote the symplectic form on

ˆ

**Twisted Holomorphic Symplectic**

**Forms**

**1.1**

**Introduction**

This chapter is basically the content of [Is16], in which we are concerned with compact complex manifolds which admit a particular kind of structure: holomorphic non-degenerate 2-forms valued in a line bundle. Manifolds admitting such a structure will be called twisted holomorphic symplectic (THS). The problem has diﬀerent analogues that have been intensively studied. On the one hand, there is the non-twisted problem concerning holomorphic symplectic forms. On the other hand, its symmetric avatar consists in the study of holomorphic (conformal) metrics. In the compact setting, the class of complex manifolds of Kähler type admitting holomorphic symplectic forms coincides with the class of hyperkähler manifolds, as shown in [Bea83b]. There is a rich literature concerning this subject, and its study is ongoing. Turning to the symmetric counterpart, the situation is somewhat diﬀerent. Although the class of compact Kähler manifolds admitting a holomorphic metric is rather small – they are all ﬁnitely covered by complex tori, as shown in [IKO80], as soon as one allows the structure to be twisted – thus studying holomorphic conformal structures – one enters a very rich class of manifolds. A complete classiﬁcation of these has been reached only in dimension 2 and 3, in [KO82] and [JR05].

Even though one could expect that the class of THS manifolds is also wide, it turns out that the situation is not much diﬀerent from the non-twisted case. More precisely, we show in Theorem 1.3.5 that compact THS manifolds of Kähler type are locally hyperkähler. In particular, the presence of such a structure ensures the existence of a Ricci-ﬂat Kähler metric, and with respect to the connection induced by this metric the form is parallel.

Roughly speaking, the proof goes as follows: we ﬁrst notice that the THS form induces local Lefschetz-type operators acting on the sheaves of holomorphic forms Ω∗, which then determine a local splitting of Ω3 into Ω1 and some other summand. This, in turn, allows us to ﬁnd local holomorphic 1-forms which behave like connection forms on the line bundle where the twisted form takes its values. Finally, this means that the bundle admits a holomorphic connection, thus also a ﬂat one, and that the manifold is Ricci-ﬂat locally holomorphic symplectic, thus locally hyperkähler.

In the next section, we give a more precise description of THS manifolds. In Theorem 1.4.1 1

2 *1.2. Holomorphic symplectic manifolds*

we show that they are ﬁnite cyclic quotients of hyperkähler manifolds. Then we investigate under which conditions a locally hyperkähler manifold admits a THS form. The two classes do not coincide, and this is essentially because locally hyperkähler manifolds behave well on products, while THS manifolds never do, as shown in Corollary 1.4.2. Still, for locally irreducible manifolds, the two classes coincide by Corollary 1.4.3. Finally, for the intermediate case of irreducible, locally reducible manifolds, a discussion depending on the compactness of the universal cover is done in the remaining part of Section 1.4. As a consequence, we also obtain that strict THS manifolds with ﬁnite fundamental group are necessarily projective.

**1.2**

**Holomorphic symplectic manifolds**

We start by discussing the complex symplectic case. For this, let us ﬁrst deﬁne the objects we will be interested in:

**Definition 1.2.1: A Riemannian manifold (M, g) is called hyperkähler if it admits three***complex structures I, J and K which:*

1. are compatible with the metric, i.e.

*g( _{·, ·) = g(I·, I·) = g(J·, J·) = g(K·, K·)}*

2. verify the quaternionic relations:

*IJ =−JI = K*

*3. are parallel with respect to the Levi-Civita connection given by g.*

In particular, a hyperkähler manifold is Kähler with respect to its ﬁxed metric and any complex
*structure aI + bJ + cK, with a, b and c real constants verifying a*2*+ b*2*+ c*2 = 1.

*Equivalently, we could say that a 4n-dimensional Riemannian manifold (M, g) is hyperkähler*
*iﬀ its holonomy group is a subgroup of Sp(n).*

**Definition 1.2.2: A holomorphic 2-form on a complex manifold M , ω***∈ H*0* _{(M, Ω}*2

*M*), is

*called a holomorphic symplectic form if it is nondegenerate in the following sense:*

*ιvωx*= 0*⇒ v = 0, ∀x ∈ M, ∀v ∈ Tx1,0M,*

*where ιv* *is the contraction with v.*

*We call a manifold admitting such a form a holomorphic symplectic manifold.*

*In particular, a holomorphic symplectic manifold (M, ω) has even complex dimension 2m and*

*ωm* *is a nowhere vanishing holomorphic section of the canonical bundle KM*=det Ω1*M*. Thus,

*KM* *is holomorphically trivial and c*1*(M ) = 0.*

*It can be easily seen that, once we ﬁx a complex structure on a hyperkähler manifold M , say*

*I, there exists a holomorphic symplectic form ω on (M, I) deﬁned by:*
*ω( _{·, ·) = g(J·, ·) + ig(K·, ·)}*

Thus, a hyperkähler manifold is a holomorphic symplectic manifold (but not in a canonical way). In the compact case, the converse is also true:

**Theorem 1.2.3: (Beauville, [Bea83b]) Let (M, I) be a compact complex manifold of Kähler**

*type admitting a holomorphic symplectic form. Then, for any Kähler class α* * _{∈ H}*2

*(M, R),*

*there exists a unique metric g on M which is Kähler with respect to I, representing α, so that*

*(M, g) is hyperkähler.*

*Moreover, the manifold (M, I) admits a metric with holonomy exactly Sp(m) if and only if it*
*is simply connected and admits a unique holomorphic symplectic form up to multiplication by*
*a scalar.*

**Remark 1.2.4: The existence and uniqueness of the Kähler metric representing the given**
Kähler class comes from Yau’s theorem: it is exactly the unique representative in the class
that has vanishing Ricci curvature. Consequently, the holomorphic symplectic form in the
theorem is parallel with respect to the Levi-Civita connection given by this Ricci-ﬂat metric.

**1.3**

**Twisted holomorphic symplectic manifolds**

We will now concentrate on the twisted case, and see that the situation is similar to the non-twisted one. Speciﬁcally, we will show that a Kähler manifold admitting a non-degenerate twisted holomorphic form admits a locally hyperkähler metric which is moreover Kähler for the given complex structure. With respect to the connection induced by this metric, the form will be parallel.

**Definition 1.3.1:** *A Riemannian manifold (M4m, g) is called locally hyperkähler if its*

universal cover with the pullback metric is hyperkähler or, equivalently, if the restricted
holonomy group Hol0*(g) is a subgroup of Sp(m). If, moreover, the manifold admits a global*
*complex structure I which is parallel with respect to the Levi-Civita connection induced by g,*
*we will call it Kähler locally hyperkähler, or KLH for short.*

Hence, a locally hyperkähler manifold is one which admits locally three orthogonal complex
structures parallel for the Levi-Civita connection and verifying the quaternionic relations. It
*can be shown that in the case of a KLH manifold (M, g, I), one of these complex structures*
*can be taken to be I, so that an equivalent deﬁnition for KLH is a Kähler manifold which*
*admits two local parallel complex structures preserved by g which verify the quaternionic*
*relations together with I.*

**Definition 1.3.2:** *Let (M, I) be a compact complex manifold, let L be a holomorphic line*
*bundle over M . A non-degenerate L-valued holomorphic form*

*ω _{∈ H}*0

*(M, Ω*2

_{M}

_{⊗ L)}*is called a twisted holomorphic symplectic form, or THS, and also the manifold with the*
*endowed structure (M, I, L, ω) is called a THS manifold.*

**Remark 1.3.3: Like in the symplectic setting, the existence of a THS form implies that M***is of even complex dimension 2m. Moreover, ωm* is a nowhere vanishing holomorphic section
*of the line bundle KM* *⊗ Lm. Thus, we have a holomorphic isomorphism Lm* ∼*= KM*∗ . In

*particular, any metric on M naturally induces one on L, and we also have*

4 *1.3. Twisted holomorphic symplectic manifolds*

**Remark 1.3.4: Any complex surface M , of Kähler type or not, is THS in a tautological***way. Simply take L to be K _{M}*∗ , so that Ω2

_{M}_{⊗ L = K}M*⊗ KM*∗ is holomorphically trivial. Thus,

any non-zero section of this bundle is a twisted-symplectic form, which in local holomorphic
*coordinates (z, w) on M is of the form*

*ω = λdz∧ dw ⊗* *∂*

*∂z* ∧
*∂*

*∂w,* *λ∈ C.*

Therefore, the class of THS manifolds is interesting only starting from complex dimension 4. Our main result in this section is the following:

**Theorem 1.3.5:** *Let (M2m, I, L, ω), m > 1, be a compact THS manifold of Kähler type, and*
*let α _{∈ H}*2

_{(M, R) be a Kähler class. Then there exists a unique Kähler metric g with respect}*to I representing α so that (M, g, I) is KLH. Moreover, L is unitary flat and ω is parallel with*
*respect to the natural connection induced by g on L.*

*Proof. Let* * _{{U}i*}

*i*

*be a trivializing open cover for the line bundle L and for each i, let σi*∈

*H*0*(Ui, L) be a holomorphic frame, so that the holomorphic transition functions{gij*}*ij* are

*given by σi* *= gijσj. Then, if we write over Ui*

*ω = ωi⊗ σi*

*we get local holomorphic symplectic forms ωi* *that verify, on Ui∩ Uj, ωi* *= gjiωj*.

*The ωi*’s, being holomorphic, induce the morphisms of sheaves ofO*Ui-modules over Ui*:

*Lk*: Ω*kUi* → Ω

*k+2*
*Ui*

*Lkα = ωi∧ α.*

**Lemma 1.3.6: For m > 1 we have an isomorphism of sheaves of**_{O}*Ui-modules:*

Ω3* _{U}_{i}* ∼= Ω1

*Ui*⊕ Ω

3
*0,Ui*

*where Ω*3_{0,U}_{i}*is the sheaf Ker(Lm−2*_{3} : Ω3* _{U}_{i}* → Ω

*n−1Ui*

*) and n = 2m.*

*Proof. We claim that Lm−1*_{1} : Ω1* _{U}_{i}* → Ω

*n−1Ui*

*is an isomorphism of sheaves over Ui*. We inspect

*this at the germ level, so we ﬁx z* * _{∈ U}i*. Since the corresponding free O

*z*-modules have the

*same rank, it suﬃces to prove the injectivity of Lm−1 _{1,z}* . But this becomes a trivial linear algebra

*problem, noting that we can always ﬁnd a basis over C in T1,0*∗

_{M}*z* *{e*1*, . . . , em, f*1*, . . . , fm*} so
that
*ωi(z) =*
*m*
X
*s=1*
*es∧ fs.*

*Next, since Lm−1*_{1} *= Lm−2*_{3} * _{◦ L}*1

*we get that L*1

*is injective and Lm−2*3 is surjective. Hence, we have an exact sequence of sheaves:

0 //Ω3
*0,Ui*
//Ω3
*Ui*
*T* _{//}
Ω1
*Ui*
//0
*where T := (Lm−1*_{1} )−1_{◦ L}m−2

3 *. But T admits as a section L*1: Ω1*Ui* → Ω

3

*Ui, as T L*1 = id. Thus,

*Now, we have dωi*∈ Ω3* _{M}(Ui*), so we can write:

*dωi* *= ωi∧ θi+ ξi* (1.3.1)

*with θi* ∈ Ω1*M(Ui) and ξi*∈ Ω3*0,M(Ui*) holomorphic sections uniquely determined by the previous

*lemma. Since ωi= gjiωj*, we get:

*dgji∧ ωj+ gjidωj* *= gjiωj∧ θi+ ξi*
whence
*ωj∧ θj+ ξj* *= dωj* *= ωj* *∧ θi*+
1
*gji*
*ξi*−
*dgji*
*gji* *∧ ωj*

*Thus, applying again the previous lemma, we obtain that the θi*’s change by the rule:

*θi* *= θj+ d log gji.* (1.3.2)

*Hence, the diﬀerential operator D : C*∞*(M, L)→ C*∞* _{(M, T}*∗

_{M}_{⊗ L) given over U}*i* by

*D(f* * _{⊗ σ}i) = (df− θi*)

*⊗ σi*

*is a well deﬁned connection on L. On the other hand, given some Hermitian metric h on L,*
*its Chern connection Dh* *must diﬀer from D by a linear operator:*

*Dh* *= D + A,* *A _{∈ C}*∞

*(T*∗

*M*

_{⊗ EndL).}*Moreover, since D0,1* *= (Dh*)*0,1*= ¯*∂L, A must be a global (1, 0)-form on M .*

*Now Θ(Dh) = Θ(D) + dA, and since Θ(D)Ui* =*−dθi* *is of type (2,0) and iΘ(D*

*h*_{) is a real}

*(1,1)-form, we have that iΘ(Dh _{) = i ¯}_{∂A is exact in H}1,1_{(M, R). But on a compact Kähler}*

*manifold H1,1(M, R)⊂ H*2

*dR(M, R), so 2πc*1*(L) = [iΘ(Dh*)] = 0*∈ HdR*2 *(M, R).*

*Thus we also get c*1*(M ) = mc*1*(L) = 0. So, by Yau’s theorem, there exists a unique Ricci-ﬂat*
*Kähler metric g whose fundamental form ωg* *represents the given class α.*

Now, on the sections of_{E}* _{M}2,0_{⊗ L we have the Weitzenböck formula (see for instance [M]):}*
2 ¯

*∂*∗

*∂ =*¯

_{∇}∗

_{∇ + R}

where* _{∇ is the naturally induced connection by g on E}_{M}2,0_{⊗ L and R is a curvature operator}*
which on decomposable sections is given by:

*R(β ⊗ s) = iρgβ⊗ s + β ⊗ Trωg(iΘ(L))s*

*with ρg* :E* _{M}2,0*→ E

*the induced action of the Ricci form onE*

_{M}2,0

_{M}2,0. Now, since g is Ricci-ﬂat,*ρg* *≡ 0. Also, if we consider the curvatures induced by g, we have:*

0 =*−iρ = Θ(KM*∗ *) = Θ(Lm*)

*so the induced connection on L is ﬂat and* _{R vanishes.}

*Hence, applying the Weitzenböck formula to ω, we get 0 =*_{∇}∗_{∇ω or also, after integrating}*over M , _{k∇ωk}*2

6 *1.3. Twisted holomorphic symplectic manifolds*

*Finally, if we let π : ( ˜M , ˜g, ˜I)→ (M, g, I) be the universal cover with the pullback metric and*
*complex structure, we have that π*∗*L is holomorphically trivial and ˜ω = π*∗*ω* * _{∈ H}*0( ˜

*M , Ω*2

_{M}_{˜}) is a holomorphic symplectic form. By the Cheeger-Gromoll theorem, ˜

*M ∼*= C

*l× M*0, where

*M*0 is compact, simply connected, Kähler, Ricci-ﬂat, and C*l* has the standard Kähler metric.
*Moreover, by the theorems of de Rham and Berger, the holonomy of M*0 is a product of groups
*of type Sp(k) and SU(k). We have that ˜ω is a parallel section of*

V2* _{T}*∗

_{M =}_{˜}V2

_{pr}∗

1*T*∗C*l*⊕ (pr∗1*T*∗C*l*⊗ pr∗2*T*∗*M*0)⊕V2pr∗2*T*∗*M*0
But pr∗_{1}*T*∗C*l*_{⊗ pr}∗

2*T*∗*M*0 ∼*= (T*∗*M*0)*⊕l* has no parallel sections by the holonomy principle, so
˜

*ω is of the form ωc+ ω*0*, with ωc, ω*0 holomorphic symplectic forms on C*l, M*0 respectively.
*Thus, l is even, so Cl* *is hyperkähler, and also, by Theorem 1.3, M*0 is hyperkähler. It follows
*that (M, g, I) is KLH.*

This concludes the proof of the theorem.

**Remark 1.3.7: Note that D**1,0*is actually a holomorphic connection on L, so this gives*
*another reason of why L must be unitary ﬂat.*

**Remark 1.3.8: The ﬂat connection induced by g on L does not depend on the Kähler class**

*α. It is uniquely determined by ω and is equal to the connection D given in the above proof.*

*To see this, let Dg* _{be the Chern connection on L induced by g and write D}g_{σ}

*i= τi⊗ σi*. Then
we have:
0 =_{∇ω = ∇ω}i⊗ σi+ ωi⊗ τi⊗ σi.*So, denoting by a :*E*M2,0⊗ (T*∗*M* *⊗ C) ⊗ L → (E*
*3,0*
*M* ⊕ E
*2,1*

*M* )*⊗ L the antisymmetrization map,*

we get:

*dωi= a(∇ωi*) =*−ωi∧ τi.*

*Thus, by (1.3.1) we deduce that ξi= 0 and τi* =*−θi, i.e. Dg* *= D.*

**Remark 1.3.9:** *If we only suppose that ω is a non degenerate (2, 0) twisted form, not*
*necessarily holomorphic, then ω still induces a connection on L in the same manner. This*
time, we have the morphisms of sheaves of _{E}*Ui-modules Lk* : E

*k,0*
*Ui* → E
*k+2,0*
*Ui* which induce
isomorphisms_{E}* _{M}3,0(Ui*) ∼=E

*)⊕ E*

_{M}1,0(Ui

_{0,M}3,0*(Ui*). Writing E

*M3,0(Ui*)

*∋ ∂ωi*

*= ωi∧ θi+ ξi,*

*we get the (1, 0)-forms θi* *which deﬁne a connection D just as before. It is only at this point*

*that the holomorphicity of ω becomes essential in order to have that D deﬁnes a holomorphic*
*connection on L.*

Actually, the complex manifolds which admit a non degenerate (2,0)-form valued in a complex
*line bundle are exactly those which have a topological Sp(m)U(1) structure. As expected,*
these are not necessarily locally hyperkähler: a counterexample is given by the quadric
Q6 *=SO(7)/U(3)*⊂ P7C, which is a Kähler manifold with topological Sp(3)U(1) structure, see
[MPS13], but is not KLH, since it has positive ﬁrst Chern class.

**Remark 1.3.10: Note that the Kähler hypothesis was heavily used during the proof. So one**
could ask two questions in the non-Kähler setting:

(1) Does it follow that a compact complex THS manifold has holomorphic torsion canonical
*bundle, so that ω determines a holomorphic symplectic form on some ﬁnite unramiﬁed cover*
*of M ?*

(2) Which are the compact complex manifolds admitting a holomorphic symplectic form?
For the ﬁrst question, the problem comes from cohomology. For a general compact complex
manifold one can deﬁne many cohomologies (de Rham, Dolbeault, Bott-Chern, Aeppli) which
*are not necessarily comparable. In particular, one does not always have a map from H1,1*_{¯}

*∂* *(M, C)*

*to H _{dR}*2

*(M, C). Since what we actually show is that c*1

*(KM*)

*¯= 0, we cannot conclude that this Chern class vanishes in all other cohomologies (except for Aeppli). Moreover, even if it*

_{∂}*was the case, this would still not imply that KM*is holomorphically torsion, see [To15] for a

detailed discussion and for examples showing the nonequivalence of the notions.

For Fujiki’s class*C manifolds, the answer is yes though. Since these manifolds satisfy the ∂ ¯*

∂-lemma, we can conclude that the ﬁrst Chern class of the manifold vanishes in all cohomologies.
We then use the result of [To15] stating that a Fujiki’s class_{C manifold M with Bott-Chern}*class c*1*(M )BC* = 0 has holomorphic torsion canonical bundle.

*Examples of THS manifolds which do not verify the ∂ ¯∂-lemma can be given as follows: let*
*S be a primary Kodaira surface. It admits a closed holomorphic symplectic form, thus also*
*Sm* _{does. Moreover, if Γ =< γ >}_{⊂Aut(S) is a ﬁnite cyclic group so that S/}

Γ is a secondary
*Kodaira surface, then Sm/<γ,...,γ>* is THS by Theorem 1.4.1 in the next section. Note that

this manifold still has holomorphic torsion canonical bundle.

Regarding the second question, what we can say for sure is that the holomorphic symplectic class
strictly contains the hyperkähler manifolds. Compact non-Kähler manifolds with holomorphic
symplectic forms were constructed by Guan and Bogomolov (see [Gu94] and [Bo96]). A
non-Kähler example with non-closed non-degenerate holomorphic form is given as follows:
start with a global complex contact manifold, the Iwasawa 3-fold for instance, that is a complex
*manifold M2m+1* *admitting a global holomorphic form η* *∈ H*0* _{(M, Ω}*1

*M) such that η∧ dηm* is

**nowhere zero. Let T be a 1-dimensional complex torus, and take on X = M**_{× T the form}

*ω = dη + θ∧ η, where θ is a generator of H*0*_{(T, Ω}*1

**T***). Then ω is holomorphic symplectic and*
veriﬁes 0*6= dω = θ ∧ ω. More examples can be constructed as complex mapping tori over M:*
*let f* * _{∈Aut(M, η) be a contactomorphism, i.e. f}*∗

_{η = η. Write T = C/}Λ, Λ = Z*⊕ τZ, and let*
*Λ act on M by 1.x = x and τ.x = f (x). Then ω descends to Mf* *:= M*×ΛC, which is again
holomorphic symplectic. These examples are the holomorphic version of what is usually called
locally conformally symplectic manifolds.

**1.4**

**A characterization**

In this section, we want to investigate the converse problem. It is not true that all KLH
manifolds are twisted holomorphic symplectic. Already we will see that a product of strictly
THS manifolds is never THS, but it turns out that being reducible is not the only obstruction.
In what follows, we will give some description of THS manifolds and their fundamental groups.
By a strictly twisted holomorphic symplectic manifold we always mean a THS manifold
*(M, I, L, ω) such that the line bundle L is not holomorphically trivial.*

**Theorem 1.4.1:** *A compact Kähler manifold M of complex dimension > 2 is THS if and*
*only if there exists a holomorphic symplectic form ω*0 *on its universal cover ˜M so that the*

8 *1.4. A characterization*

*action of Γ = π*1*(M ) on H*0( ˜*M , Ω*2* _{M}*˜

*) preserves Cω*0

*. In particular, any THS manifold is a*

*finite cyclic quotient of a hyperkähler manifold.*

*Proof. Suppose ﬁrst that M admits a twisted-symplectic form*
*ω∈ H*0*(M, Ω*2_{M}⊗ L).

*Then, by Theorem 1.3.5, L is unitary ﬂat, and thus given by a unitary representation*

*ρ : Γ _{→U(1), i.e. if we see π : ˜}M*

_{→ M as a Γ-principal bundle over M, we have L = ˜}M_{×}

*ρ*C.

*Let si* *: Ui* → ˜*M be local sections of π : ˜M* *→ M over a trivializing cover {Ui*}. We then

*have si= γijsj* *on Ui∩ Uj, where γij* *: Ui∩ Uj* → Γ are the transition functions for ˜*M . Then,*

*σi:= [si, 1] are local frames for L, where [·, ·] denotes the orbit of an element of ˜M* × C under

*the left action of Γ. The locally constant functions gij* *:= ρ(γij*−1) are the transition functions

*for L verifying*

*σi= [γijsj, 1] = [sj, ρ(γij*−1*)] = gijσj.*

*Since π*∗_{L is trivial, there exist f}

*i* ∈ O∗* _{M}*˜

*(π*−1

*Ui) such that π*∗

*gij*=

_{f}f_{j}i*on π*−1

*Ui∩ π*−1

*Uj*. Also,

the sections *π*∗*σi*

*fi* *∈ H*

0* _{(π}*−1

_{U}*i, π*∗*L) all coincide on intersections and are non vanishing, thus*

*giving a global frame for π*∗*L which we can suppose equal to 1, so that π*∗*σi* *= fi*. Thus, if we

*write ω = ωi⊗ σi* *and deﬁne ω*0*:= π*∗*ω, we get:*

*ω*0|*π*−_{1}

*Ui* *= π*

∗_{ω}

*ifi*

*and, for any γ* _{∈ Γ:}

*γ*∗*ω*0|*π*−_{1}
*Ui* *= π*
∗_{ω}*iγ*∗*fi* *= ω*0
*γ*∗*fi*
*fi*

*Moreover, for any γ, we have on π*−1*Ui∩ π*−1*Uj* :

*fj*

*fi*

= *fj* *◦ γ*

*fi◦ γ* *⇔ gij◦ π = gij◦ π ◦ γ*

hence the constant function *fi◦γ*

*fi* *does not depend on i.*

On the other hand, we have:

*γ*∗*fi*
*fi*
= *[si◦ π ◦ γ, 1]*
*[si◦ π, 1]*
= *[si◦ π, ρ(γ*
−1_{)]}
*[si◦ π, 1]*
= 1
*ρ(γ)* (1.4.1)

*Hence Γ preserves the subspace Cω*0 *⊂ H*0( ˜*M , Ω*2* _{M}*˜

*) and ρ is determined by the action of Γ on*

*the holomorphic symplectic form ω*0 by:

1

*ρ(γ)· ω*0*= γ*

∗* _{ω}*
0

*.*

*Conversely, suppose a holomorphic symplectic form ω*0 is an eigenvector for Γ acting on

*H*0_{( ˜}* _{M , Ω}*2
˜

*M*). Deﬁne

*ρ : Γ*

_{→ C}∗

*γ*

_{7→}

*ω*0

*γ*∗

*0*

_{ω}